library(tidyverse)
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library(astsa)
library(vars)
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library(fpp)
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library(ggplot2)
#setwd("/Users/duleepa/Desktop/Time_Series/Project")
trip_data = readRDS("Monthly_Trip.rds")
head(trip_data)
ports <- trip_data %>%
filter(Port_Name %in% c("Sault Sainte Marie","Detroit","Port Huron", "Buffalo Niagara Falls", "Alexandria Bay", "Ogdensburg", "Massena", "Champlain Rouses Point"))
dim(ports)
## [1] 29343 5
S_S_Marie <- ports %>%
filter(Port_Name %in% c("Sault Sainte Marie")) %>%
filter(!grepl("Passengers", Measure, ignore.case = TRUE)) %>%
group_by(Year, Month) %>%
summarise(Total = sum(Value))
## `summarise()` has grouped output by 'Year'. You can override using the
## `.groups` argument.
Detroit <- ports %>%
filter(Port_Name %in% c("Detroit")) %>%
filter(!grepl("Passengers", Measure, ignore.case = TRUE)) %>%
group_by(Year, Month) %>%
summarise(Total = sum(Value))
## `summarise()` has grouped output by 'Year'. You can override using the
## `.groups` argument.
Port_Huron <- ports %>%
filter(Port_Name %in% c("Port Huron")) %>%
filter(!grepl("Passengers", Measure, ignore.case = TRUE)) %>%
group_by(Year, Month) %>%
summarise(Total = sum(Value))
## `summarise()` has grouped output by 'Year'. You can override using the
## `.groups` argument.
B_Niagara <- ports %>%
filter(Port_Name %in% c("Buffalo Niagara Falls")) %>%
filter(!grepl("Passengers", Measure, ignore.case = TRUE)) %>%
group_by(Year, Month) %>%
summarise(Total = sum(Value))
## `summarise()` has grouped output by 'Year'. You can override using the
## `.groups` argument.
Alexandria <- ports %>%
filter(Port_Name %in% c("Alexandria Bay")) %>%
filter(!grepl("Passengers", Measure, ignore.case = TRUE)) %>%
group_by(Year, Month) %>%
summarise(Total = sum(Value))
## `summarise()` has grouped output by 'Year'. You can override using the
## `.groups` argument.
Ogdensburg <- ports %>%
filter(Port_Name %in% c("Ogdensburg")) %>%
filter(!grepl("Passengers", Measure, ignore.case = TRUE)) %>%
group_by(Year, Month) %>%
summarise(Total = sum(Value))
## `summarise()` has grouped output by 'Year'. You can override using the
## `.groups` argument.
Massena <- ports %>%
filter(Port_Name %in% c("Massena")) %>%
filter(!grepl("Passengers", Measure, ignore.case = TRUE)) %>%
group_by(Year, Month) %>%
summarise(Total = sum(Value))
## `summarise()` has grouped output by 'Year'. You can override using the
## `.groups` argument.
Champlain_Point <- ports %>%
filter(Port_Name %in% c("Champlain Rouses Point")) %>%
filter(!grepl("Passengers", Measure, ignore.case = TRUE)) %>%
group_by(Year, Month) %>%
summarise(Total = sum(Value))
## `summarise()` has grouped output by 'Year'. You can override using the
## `.groups` argument.
S_S_Marie_ts <- ts(S_S_Marie$Total, frequency = 12, start = c(1996, 4))
Detroit_ts <- ts(Detroit$Total, frequency = 12, start = c(1996, 4))
Port_Huron_ts <- ts(Port_Huron$Total, frequency = 12, start = c(1996, 4))
B_Niagara_ts <- ts(B_Niagara$Total, frequency = 12, start = c(1996, 4))
Alexandria_ts <- ts(Alexandria$Total, frequency = 12, start = c(1996, 4))
Ogdensburg_ts <- ts(Ogdensburg$Total, frequency = 12, start = c(1996, 4))
Massena_ts <- ts(Massena$Total, frequency = 12, start = c(1996, 4))
Champlain_Point_ts <- ts(Champlain_Point$Total, frequency = 12, start = c(1996, 4))
# Plot the first time series
tsplot(S_S_Marie_ts, col = "blue", ylim = c(0, max(c(S_S_Marie_ts, Detroit_ts, Port_Huron_ts))), ylab = "Volume")
# Add the second and third time series to the plot
lines(Detroit_ts, col = "red")
lines(Port_Huron_ts, col = "green")
# Add legend
legend("topright", legend = c("Sault Ste. Marie", "Detroit", "Port Huron"), col = c("blue", "red", "green"), lty = 1)
#Plot
tsplot(B_Niagara_ts, col = "blue", ylim = c(0, max(c(S_S_Marie_ts, Detroit_ts, Port_Huron_ts))), ylab = "Volume")
# Add lines with different colors for all ports
lines(Detroit_ts, col = "green")
lines(S_S_Marie_ts, col = "purple")
lines(Port_Huron_ts, col = "orange")
lines(Alexandria_ts, col = "red")
lines(Ogdensburg_ts, col = "yellow")
lines(Massena_ts, col = "cyan")
lines(Champlain_Point_ts, col = "magenta")
# Add legend
legend("topright", legend = c("Sault Ste. Marie", "Detroit", "Port Huron", "Alexandria", "Ogdensburg", "Massena", "Champlain Point"),
col = c("blue", "green", "purple", "orange", "red", "yellow", "cyan", "magenta"), lty = 1)
Detroit_modes <- ports %>%
filter(Port_Name %in% c("Detroit")) %>%
group_by(Year, Month, Measure) %>%
summarise(Total = sum(Value)) %>%
filter(!grepl("Passengers", Measure, ignore.case = TRUE))
## `summarise()` has grouped output by 'Year', 'Month'. You can override using the
## `.groups` argument.
#passenger
Detroit_personal_vehicles <- Detroit_modes %>%
filter(Measure %in% "Personal Vehicles")
Detroit_buses <- Detroit_modes %>%
filter(Measure %in% "Buses")
Detroit_trains <- Detroit_modes %>%
filter(Measure %in% "Trains")
Detroit_personal_vehicles_ts <- ts(Detroit_personal_vehicles$Total, frequency = 12, start = c(1996, 4))
Detroit_buses_ts <- ts(Detroit_buses$Total, frequency = 12, start = c(1996, 4))
Detroit_trains_ts <- ts(Detroit_trains$Total, frequency = 12, start = c(1996, 4))
#freight
Detroit_Trucks <- Detroit_modes %>%
filter(Measure %in% "Trucks")
Detroit_Truck_Loaded <- Detroit_modes %>%
filter(Measure %in% "Truck Containers Loaded")
Detroit_Truck_Empty <- Detroit_modes %>%
filter(Measure %in% "Truck Containers Empty")
Detroit_rail <- Detroit_modes %>%
filter(grepl("Rail", Measure, ignore.case = TRUE))
Detroit_Trucks_ts <- ts(Detroit_Trucks$Total, frequency = 12, start = c(1996, 4))
Detroit_Truck_Loaded_ts <- ts(Detroit_Truck_Loaded$Total, frequency = 12, start = c(1996, 4))
Detroit_rail_ts <- ts(Detroit_rail$Total, frequency = 12, start = c(1996, 4))
# Plot the first time series
tsplot(Detroit_personal_vehicles_ts, col = "blue", ylim = c(0, max(c(Detroit_personal_vehicles_ts, Detroit_buses_ts, Detroit_trains_ts))), ylab = "Volume")
# Add the second and third time series to the plot
lines(Detroit_buses_ts, col = "red")
lines(Port_Huron_ts, col = "green")
# Add legend
legend("topright", legend = c("Detroit personal vehicles", "Detroit buses", "Detroit trains"), col = c("blue", "red", "green"), lty = 1)
# Plot the first time series
tsplot(Detroit_Trucks_ts, col = "blue", ylim = c(0, max(c(Detroit_Trucks_ts, Detroit_rail_ts))), ylab = "Volume")
# Add the second and third time series to the plot
lines(Detroit_rail_ts, col = "red")
# Add legend
legend("topright", legend = c("Detroit Trucks", "Detroit rail"), col = c("blue", "red", "green"), lty = 1)
multivariate_trip <- cbind(S_S_Marie, Detroit[,3], Port_Huron[,3], B_Niagara[,3], Alexandria[,3], Ogdensburg[,3], Massena[,3], Champlain_Point[,3])
## New names:
## • `Total` -> `Total...3`
## • `Total` -> `Total...4`
## • `Total` -> `Total...5`
## • `Total` -> `Total...6`
## • `Total` -> `Total...7`
## • `Total` -> `Total...8`
## • `Total` -> `Total...9`
## • `Total` -> `Total...10`
colnames(multivariate_trip)[3] <- "S_S_Marie"
colnames(multivariate_trip)[4] <- "Detroit"
colnames(multivariate_trip)[5] <- "Port_Huron"
colnames(multivariate_trip)[6] <- "B_Niagara"
colnames(multivariate_trip)[7] <- "Alexandria"
colnames(multivariate_trip)[8] <- "Ogdensburg"
colnames(multivariate_trip)[9] <- "Massena"
colnames(multivariate_trip)[10] <- "Champlain_Point"
multivariate_ts <- ts(multivariate_trip[3:10], frequency = 12, start = c(1996, 4))
class(multivariate_ts)
## [1] "mts" "ts" "matrix"
plot(multivariate_ts)
theme_set(theme_bw())
autoplot(multivariate_ts) +
ggtitle("Port Time-Series") +
ylab("Volume") +
theme(plot.title = element_text(hjust = 0.5))
## Testing for stationarity
library(tidyverse)
library(lmtest)
adf_results <- apply(multivariate_ts, 2, function(x) adf.test(x)$p.value)
# p-values based on Port
(result_table <- tibble(Port = names(adf_results), p_value = adf_results))
library(fGarch)
## NOTE: Packages 'fBasics', 'timeDate', and 'timeSeries' are no longer
## attached to the search() path when 'fGarch' is attached.
##
## If needed attach them yourself in your R script by e.g.,
## require("timeSeries")
library(MTS)
##
## Attaching package: 'MTS'
## The following object is masked from 'package:vars':
##
## VAR
# Differencing the whole mts
stnry = diffM(multivariate_ts) #difference operation on a vector of time series. Default order of differencing is 1.
# Retest
adf_stnry_results <- apply(stnry, 2, function(x) adf.test(x)$p.value)
## Warning in adf.test(x): p-value smaller than printed p-value
## Warning in adf.test(x): p-value smaller than printed p-value
## Warning in adf.test(x): p-value smaller than printed p-value
## Warning in adf.test(x): p-value smaller than printed p-value
## Warning in adf.test(x): p-value smaller than printed p-value
## Warning in adf.test(x): p-value smaller than printed p-value
## Warning in adf.test(x): p-value smaller than printed p-value
## Warning in adf.test(x): p-value smaller than printed p-value
# p-values based on Port
(result_table <- tibble(Port = names(adf_stnry_results), p_value = adf_stnry_results))
## VAR modeling
plot.ts(stnry, main = "Stationary Time Series")
autoplot(ts(stnry,
start = c(1996,4),
frequency = 12)) +
ylab("Volume") +
ggtitle("Stationary Time Series")
stnry_ts <- ts(stnry, frequency = 12, start = c(1996, 4))
VARselect(stnry_ts, lag.max=8, type="none")
## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 7 4 2 7
##
## $criteria
## 1 2 3 4 5
## AIC(n) 1.547234e+02 1.538456e+02 1.530843e+02 1.526804e+02 1.525921e+02
## HQ(n) 1.550303e+02 1.544593e+02 1.540049e+02 1.539078e+02 1.541264e+02
## SC(n) 1.554912e+02 1.553812e+02 1.553877e+02 1.557515e+02 1.564311e+02
## FPE(n) 1.568793e+67 6.525513e+66 3.053103e+66 2.045282e+66 1.882973e+66
## 6 7 8
## AIC(n) 1.525190e+02 1.522917e+02 1.523483e+02
## HQ(n) 1.543602e+02 1.544398e+02 1.548032e+02
## SC(n) 1.571258e+02 1.576663e+02 1.584907e+02
## FPE(n) 1.764932e+66 1.422735e+66 1.529442e+66
# Creating a VAR model with vars
var.a <- vars::VAR(stnry,
lag.max = 10, season = 12, #highest lag order for lag length selection according to the choosen ic
ic = "AIC", #information criterion
type = "none") #type of deterministic regressors to include
summary(var.a)
##
## VAR Estimation Results:
## =========================
## Endogenous variables: S_S_Marie, Detroit, Port_Huron, B_Niagara, Alexandria, Ogdensburg, Massena, Champlain_Point
## Deterministic variables: none
## Sample size: 314
## Log Likelihood: -26823.63
## Roots of the characteristic polynomial:
## 0.8896 0.8896 0.8708 0.8708 0.8663 0.8663 0.8547 0.8547 0.8272 0.8272 0.826 0.826 0.8211 0.8211 0.8148 0.8148 0.7919 0.7919 0.7901 0.7901 0.7848 0.7848 0.7787 0.7787 0.7692 0.7692 0.7645 0.7645 0.7569 0.7569 0.7502 0.7502 0.7423 0.7423 0.7236 0.7236 0.6925 0.6925 0.6416 0.6416 0.6392 0.6392 0.5378 0.5378 0.5129 0.5129 0.359 0.1577
## Call:
## vars::VAR(y = stnry, type = "none", season = 12L, lag.max = 10,
## ic = "AIC")
##
##
## Estimation results for equation S_S_Marie:
## ==========================================
## S_S_Marie = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + S_S_Marie.l3 + Detroit.l3 + Port_Huron.l3 + B_Niagara.l3 + Alexandria.l3 + Ogdensburg.l3 + Massena.l3 + Champlain_Point.l3 + S_S_Marie.l4 + Detroit.l4 + Port_Huron.l4 + B_Niagara.l4 + Alexandria.l4 + Ogdensburg.l4 + Massena.l4 + Champlain_Point.l4 + S_S_Marie.l5 + Detroit.l5 + Port_Huron.l5 + B_Niagara.l5 + Alexandria.l5 + Ogdensburg.l5 + Massena.l5 + Champlain_Point.l5 + S_S_Marie.l6 + Detroit.l6 + Port_Huron.l6 + B_Niagara.l6 + Alexandria.l6 + Ogdensburg.l6 + Massena.l6 + Champlain_Point.l6 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11
##
## Estimate Std. Error t value Pr(>|t|)
## S_S_Marie.l1 -5.755e-01 1.056e-01 -5.449 1.19e-07 ***
## Detroit.l1 2.490e-02 1.185e-02 2.102 0.036572 *
## Port_Huron.l1 2.166e-02 3.916e-02 0.553 0.580733
## B_Niagara.l1 -2.812e-02 1.643e-02 -1.711 0.088234 .
## Alexandria.l1 -5.094e-02 8.084e-02 -0.630 0.529149
## Ogdensburg.l1 5.165e-01 1.846e-01 2.798 0.005532 **
## Massena.l1 7.558e-02 7.429e-02 1.017 0.309965
## Champlain_Point.l1 -7.886e-02 5.862e-02 -1.345 0.179707
## S_S_Marie.l2 -4.003e-01 1.196e-01 -3.346 0.000942 ***
## Detroit.l2 1.962e-02 1.346e-02 1.458 0.146126
## Port_Huron.l2 9.186e-02 4.585e-02 2.004 0.046175 *
## B_Niagara.l2 -3.990e-02 1.984e-02 -2.011 0.045387 *
## Alexandria.l2 -1.613e-01 9.916e-02 -1.626 0.105106
## Ogdensburg.l2 2.735e-01 2.263e-01 1.209 0.227898
## Massena.l2 8.369e-02 8.650e-02 0.968 0.334173
## Champlain_Point.l2 -1.338e-01 6.979e-02 -1.918 0.056264 .
## S_S_Marie.l3 -3.137e-01 1.255e-01 -2.500 0.013064 *
## Detroit.l3 2.587e-02 1.416e-02 1.827 0.068797 .
## Port_Huron.l3 7.184e-02 5.025e-02 1.430 0.154003
## B_Niagara.l3 -5.052e-02 2.239e-02 -2.257 0.024867 *
## Alexandria.l3 -2.903e-01 1.085e-01 -2.676 0.007944 **
## Ogdensburg.l3 2.521e-01 2.510e-01 1.004 0.316282
## Massena.l3 6.892e-02 9.542e-02 0.722 0.470797
## Champlain_Point.l3 2.798e-02 7.408e-02 0.378 0.705974
## S_S_Marie.l4 -1.713e-01 1.234e-01 -1.388 0.166259
## Detroit.l4 2.517e-02 1.419e-02 1.774 0.077334 .
## Port_Huron.l4 -1.372e-03 5.002e-02 -0.027 0.978148
## B_Niagara.l4 -3.678e-02 2.239e-02 -1.643 0.101658
## Alexandria.l4 -2.538e-01 1.095e-01 -2.318 0.021265 *
## Ogdensburg.l4 4.889e-01 2.500e-01 1.956 0.051552 .
## Massena.l4 7.255e-02 9.471e-02 0.766 0.444334
## Champlain_Point.l4 -1.283e-02 7.548e-02 -0.170 0.865194
## S_S_Marie.l5 -1.729e-01 1.161e-01 -1.489 0.137849
## Detroit.l5 3.841e-02 1.337e-02 2.873 0.004408 **
## Port_Huron.l5 3.758e-02 4.679e-02 0.803 0.422600
## B_Niagara.l5 -2.247e-02 1.995e-02 -1.126 0.261049
## Alexandria.l5 -1.311e-01 1.001e-01 -1.309 0.191691
## Ogdensburg.l5 4.764e-02 2.267e-01 0.210 0.833740
## Massena.l5 1.125e-01 8.522e-02 1.320 0.187928
## Champlain_Point.l5 2.253e-02 7.222e-02 0.312 0.755274
## S_S_Marie.l6 -2.481e-02 1.025e-01 -0.242 0.808938
## Detroit.l6 1.538e-02 1.248e-02 1.232 0.219113
## Port_Huron.l6 2.425e-02 3.990e-02 0.608 0.543853
## B_Niagara.l6 -1.494e-02 1.627e-02 -0.918 0.359337
## Alexandria.l6 -1.202e-01 8.127e-02 -1.479 0.140494
## Ogdensburg.l6 5.903e-02 1.873e-01 0.315 0.752948
## Massena.l6 6.195e-02 7.266e-02 0.853 0.394720
## Champlain_Point.l6 -3.186e-02 6.051e-02 -0.526 0.599039
## sd1 3.640e+04 4.280e+03 8.503 1.58e-15 ***
## sd2 4.271e+03 5.316e+03 0.803 0.422463
## sd3 8.025e+03 4.743e+03 1.692 0.091888 .
## sd4 9.510e+03 5.596e+03 1.699 0.090491 .
## sd5 3.327e+04 4.727e+03 7.038 1.80e-11 ***
## sd6 7.828e+03 5.553e+03 1.410 0.159871
## sd7 2.462e+04 5.334e+03 4.615 6.22e-06 ***
## sd8 1.537e+04 4.568e+03 3.364 0.000887 ***
## sd9 5.955e+03 4.979e+03 1.196 0.232824
## sd10 1.699e+04 5.966e+03 2.848 0.004760 **
## sd11 2.399e+04 5.739e+03 4.179 4.02e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 8614 on 255 degrees of freedom
## Multiple R-Squared: 0.8494, Adjusted R-squared: 0.8146
## F-statistic: 24.38 on 59 and 255 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation Detroit:
## ========================================
## Detroit = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + S_S_Marie.l3 + Detroit.l3 + Port_Huron.l3 + B_Niagara.l3 + Alexandria.l3 + Ogdensburg.l3 + Massena.l3 + Champlain_Point.l3 + S_S_Marie.l4 + Detroit.l4 + Port_Huron.l4 + B_Niagara.l4 + Alexandria.l4 + Ogdensburg.l4 + Massena.l4 + Champlain_Point.l4 + S_S_Marie.l5 + Detroit.l5 + Port_Huron.l5 + B_Niagara.l5 + Alexandria.l5 + Ogdensburg.l5 + Massena.l5 + Champlain_Point.l5 + S_S_Marie.l6 + Detroit.l6 + Port_Huron.l6 + B_Niagara.l6 + Alexandria.l6 + Ogdensburg.l6 + Massena.l6 + Champlain_Point.l6 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11
##
## Estimate Std. Error t value Pr(>|t|)
## S_S_Marie.l1 3.206e-01 7.392e-01 0.434 0.66484
## Detroit.l1 -4.236e-01 8.294e-02 -5.107 6.42e-07 ***
## Port_Huron.l1 4.974e-02 2.741e-01 0.181 0.85618
## B_Niagara.l1 -3.385e-02 1.150e-01 -0.294 0.76879
## Alexandria.l1 -6.091e-01 5.658e-01 -1.077 0.28271
## Ogdensburg.l1 9.913e-01 1.292e+00 0.767 0.44365
## Massena.l1 6.027e-01 5.201e-01 1.159 0.24754
## Champlain_Point.l1 -1.090e-01 4.103e-01 -0.266 0.79066
## S_S_Marie.l2 2.788e-02 8.373e-01 0.033 0.97346
## Detroit.l2 -2.080e-01 9.419e-02 -2.208 0.02812 *
## Port_Huron.l2 -2.623e-03 3.209e-01 -0.008 0.99349
## B_Niagara.l2 -8.478e-02 1.389e-01 -0.610 0.54211
## Alexandria.l2 -8.077e-01 6.941e-01 -1.164 0.24563
## Ogdensburg.l2 -1.248e+00 1.584e+00 -0.788 0.43151
## Massena.l2 3.784e-01 6.055e-01 0.625 0.53260
## Champlain_Point.l2 -4.409e-02 4.885e-01 -0.090 0.92816
## S_S_Marie.l3 -7.068e-01 8.786e-01 -0.805 0.42183
## Detroit.l3 -1.908e-01 9.911e-02 -1.925 0.05535 .
## Port_Huron.l3 -3.833e-02 3.517e-01 -0.109 0.91331
## B_Niagara.l3 1.654e-02 1.567e-01 0.106 0.91600
## Alexandria.l3 -1.783e+00 7.594e-01 -2.347 0.01967 *
## Ogdensburg.l3 -1.654e-01 1.757e+00 -0.094 0.92509
## Massena.l3 2.441e-01 6.680e-01 0.365 0.71512
## Champlain_Point.l3 3.970e-01 5.185e-01 0.766 0.44455
## S_S_Marie.l4 2.801e-01 8.635e-01 0.324 0.74594
## Detroit.l4 2.505e-02 9.936e-02 0.252 0.80112
## Port_Huron.l4 2.895e-02 3.502e-01 0.083 0.93416
## B_Niagara.l4 -1.225e-01 1.567e-01 -0.782 0.43506
## Alexandria.l4 -1.688e+00 7.665e-01 -2.202 0.02854 *
## Ogdensburg.l4 1.909e+00 1.750e+00 1.091 0.27621
## Massena.l4 3.694e-01 6.629e-01 0.557 0.57784
## Champlain_Point.l4 -3.904e-01 5.284e-01 -0.739 0.46068
## S_S_Marie.l5 2.628e-01 8.129e-01 0.323 0.74672
## Detroit.l5 2.404e-01 9.359e-02 2.569 0.01078 *
## Port_Huron.l5 -2.134e-03 3.275e-01 -0.007 0.99481
## B_Niagara.l5 -1.834e-01 1.396e-01 -1.314 0.19012
## Alexandria.l5 -8.781e-01 7.008e-01 -1.253 0.21138
## Ogdensburg.l5 -1.632e+00 1.587e+00 -1.028 0.30484
## Massena.l5 1.727e-01 5.965e-01 0.290 0.77242
## Champlain_Point.l5 2.677e-01 5.055e-01 0.530 0.59684
## S_S_Marie.l6 1.047e-01 7.175e-01 0.146 0.88412
## Detroit.l6 -1.766e-01 8.738e-02 -2.021 0.04428 *
## Port_Huron.l6 2.534e-01 2.793e-01 0.907 0.36513
## B_Niagara.l6 -5.525e-02 1.139e-01 -0.485 0.62798
## Alexandria.l6 -1.152e+00 5.689e-01 -2.024 0.04397 *
## Ogdensburg.l6 1.399e-01 1.311e+00 0.107 0.91509
## Massena.l6 3.439e-01 5.086e-01 0.676 0.49964
## Champlain_Point.l6 1.240e-01 4.236e-01 0.293 0.76989
## sd1 1.368e+05 2.996e+04 4.567 7.71e-06 ***
## sd2 -2.886e+04 3.721e+04 -0.776 0.43863
## sd3 6.231e+04 3.320e+04 1.877 0.06169 .
## sd4 6.747e+04 3.917e+04 1.722 0.08622 .
## sd5 6.707e+04 3.309e+04 2.027 0.04373 *
## sd6 5.538e+04 3.887e+04 1.425 0.15550
## sd7 1.224e+05 3.734e+04 3.278 0.00119 **
## sd8 2.604e+04 3.198e+04 0.814 0.41617
## sd9 8.616e+04 3.485e+04 2.472 0.01408 *
## sd10 1.237e+05 4.176e+04 2.962 0.00334 **
## sd11 3.660e+03 4.017e+04 0.091 0.92749
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 60290 on 255 degrees of freedom
## Multiple R-Squared: 0.554, Adjusted R-squared: 0.4507
## F-statistic: 5.368 on 59 and 255 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation Port_Huron:
## ===========================================
## Port_Huron = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + S_S_Marie.l3 + Detroit.l3 + Port_Huron.l3 + B_Niagara.l3 + Alexandria.l3 + Ogdensburg.l3 + Massena.l3 + Champlain_Point.l3 + S_S_Marie.l4 + Detroit.l4 + Port_Huron.l4 + B_Niagara.l4 + Alexandria.l4 + Ogdensburg.l4 + Massena.l4 + Champlain_Point.l4 + S_S_Marie.l5 + Detroit.l5 + Port_Huron.l5 + B_Niagara.l5 + Alexandria.l5 + Ogdensburg.l5 + Massena.l5 + Champlain_Point.l5 + S_S_Marie.l6 + Detroit.l6 + Port_Huron.l6 + B_Niagara.l6 + Alexandria.l6 + Ogdensburg.l6 + Massena.l6 + Champlain_Point.l6 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11
##
## Estimate Std. Error t value Pr(>|t|)
## S_S_Marie.l1 -1.711e-01 2.945e-01 -0.581 0.561818
## Detroit.l1 3.870e-02 3.304e-02 1.171 0.242554
## Port_Huron.l1 -5.741e-01 1.092e-01 -5.257 3.10e-07 ***
## B_Niagara.l1 1.337e-02 4.583e-02 0.292 0.770707
## Alexandria.l1 -2.175e-01 2.254e-01 -0.965 0.335586
## Ogdensburg.l1 8.182e-01 5.147e-01 1.589 0.113189
## Massena.l1 3.562e-01 2.072e-01 1.719 0.086758 .
## Champlain_Point.l1 -2.106e-01 1.635e-01 -1.288 0.198825
## S_S_Marie.l2 -4.672e-03 3.336e-01 -0.014 0.988836
## Detroit.l2 7.574e-02 3.753e-02 2.018 0.044598 *
## Port_Huron.l2 -4.291e-01 1.279e-01 -3.356 0.000912 ***
## B_Niagara.l2 -5.783e-02 5.533e-02 -1.045 0.296901
## Alexandria.l2 -4.935e-01 2.765e-01 -1.785 0.075518 .
## Ogdensburg.l2 5.799e-01 6.311e-01 0.919 0.359050
## Massena.l2 2.577e-01 2.412e-01 1.068 0.286472
## Champlain_Point.l2 -2.335e-01 1.946e-01 -1.200 0.231308
## S_S_Marie.l3 -4.313e-01 3.500e-01 -1.232 0.219040
## Detroit.l3 5.480e-02 3.948e-02 1.388 0.166399
## Port_Huron.l3 -3.109e-01 1.401e-01 -2.219 0.027375 *
## B_Niagara.l3 -4.171e-02 6.243e-02 -0.668 0.504686
## Alexandria.l3 -7.388e-01 3.025e-01 -2.442 0.015287 *
## Ogdensburg.l3 1.014e+00 7.001e-01 1.449 0.148596
## Massena.l3 2.889e-01 2.661e-01 1.086 0.278666
## Champlain_Point.l3 1.277e-01 2.066e-01 0.618 0.536971
## S_S_Marie.l4 -1.093e-01 3.440e-01 -0.318 0.750988
## Detroit.l4 6.500e-02 3.958e-02 1.642 0.101781
## Port_Huron.l4 -2.009e-01 1.395e-01 -1.440 0.151034
## B_Niagara.l4 -8.945e-02 6.244e-02 -1.433 0.153192
## Alexandria.l4 -6.776e-01 3.054e-01 -2.219 0.027371 *
## Ogdensburg.l4 1.266e+00 6.971e-01 1.816 0.070505 .
## Massena.l4 2.121e-01 2.641e-01 0.803 0.422629
## Champlain_Point.l4 5.193e-03 2.105e-01 0.025 0.980339
## S_S_Marie.l5 -3.264e-01 3.239e-01 -1.008 0.314502
## Detroit.l5 1.464e-01 3.729e-02 3.925 0.000112 ***
## Port_Huron.l5 -6.907e-02 1.305e-01 -0.529 0.597038
## B_Niagara.l5 -9.172e-02 5.562e-02 -1.649 0.100383
## Alexandria.l5 -2.209e-01 2.792e-01 -0.791 0.429629
## Ogdensburg.l5 9.459e-02 6.323e-01 0.150 0.881199
## Massena.l5 1.583e-01 2.377e-01 0.666 0.505906
## Champlain_Point.l5 1.858e-01 2.014e-01 0.923 0.357041
## S_S_Marie.l6 -3.169e-01 2.858e-01 -1.109 0.268625
## Detroit.l6 6.501e-03 3.481e-02 0.187 0.852026
## Port_Huron.l6 -1.139e-01 1.113e-01 -1.024 0.307037
## B_Niagara.l6 -2.009e-02 4.537e-02 -0.443 0.658283
## Alexandria.l6 -3.858e-01 2.267e-01 -1.702 0.089907 .
## Ogdensburg.l6 3.426e-01 5.224e-01 0.656 0.512567
## Massena.l6 2.915e-02 2.026e-01 0.144 0.885733
## Champlain_Point.l6 2.103e-01 1.688e-01 1.246 0.213813
## sd1 8.787e+04 1.194e+04 7.361 2.51e-12 ***
## sd2 -2.464e+04 1.482e+04 -1.662 0.097676 .
## sd3 1.835e+03 1.323e+04 0.139 0.889799
## sd4 -1.840e+03 1.561e+04 -0.118 0.906221
## sd5 7.858e+04 1.318e+04 5.961 8.32e-09 ***
## sd6 1.576e+04 1.549e+04 1.017 0.309982
## sd7 5.543e+04 1.488e+04 3.726 0.000239 ***
## sd8 2.324e+04 1.274e+04 1.824 0.069294 .
## sd9 2.268e+04 1.389e+04 1.634 0.103581
## sd10 4.606e+04 1.664e+04 2.768 0.006048 **
## sd11 1.892e+04 1.601e+04 1.182 0.238212
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 24020 on 255 degrees of freedom
## Multiple R-Squared: 0.8272, Adjusted R-squared: 0.7872
## F-statistic: 20.69 on 59 and 255 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation B_Niagara:
## ==========================================
## B_Niagara = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + S_S_Marie.l3 + Detroit.l3 + Port_Huron.l3 + B_Niagara.l3 + Alexandria.l3 + Ogdensburg.l3 + Massena.l3 + Champlain_Point.l3 + S_S_Marie.l4 + Detroit.l4 + Port_Huron.l4 + B_Niagara.l4 + Alexandria.l4 + Ogdensburg.l4 + Massena.l4 + Champlain_Point.l4 + S_S_Marie.l5 + Detroit.l5 + Port_Huron.l5 + B_Niagara.l5 + Alexandria.l5 + Ogdensburg.l5 + Massena.l5 + Champlain_Point.l5 + S_S_Marie.l6 + Detroit.l6 + Port_Huron.l6 + B_Niagara.l6 + Alexandria.l6 + Ogdensburg.l6 + Massena.l6 + Champlain_Point.l6 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11
##
## Estimate Std. Error t value Pr(>|t|)
## S_S_Marie.l1 -1.873e-01 9.007e-01 -0.208 0.835452
## Detroit.l1 2.498e-01 1.011e-01 2.472 0.014087 *
## Port_Huron.l1 1.410e-01 3.340e-01 0.422 0.673334
## B_Niagara.l1 -7.735e-01 1.401e-01 -5.519 8.38e-08 ***
## Alexandria.l1 -1.005e+00 6.894e-01 -1.458 0.145958
## Ogdensburg.l1 2.802e+00 1.574e+00 1.780 0.076301 .
## Massena.l1 9.115e-01 6.337e-01 1.439 0.151513
## Champlain_Point.l1 -2.019e-01 4.999e-01 -0.404 0.686591
## S_S_Marie.l2 2.012e-01 1.020e+00 0.197 0.843804
## Detroit.l2 3.022e-01 1.148e-01 2.633 0.008976 **
## Port_Huron.l2 4.977e-01 3.910e-01 1.273 0.204278
## B_Niagara.l2 -9.207e-01 1.692e-01 -5.441 1.24e-07 ***
## Alexandria.l2 -1.305e+00 8.457e-01 -1.543 0.124058
## Ogdensburg.l2 9.411e-01 1.930e+00 0.488 0.626260
## Massena.l2 3.420e-01 7.377e-01 0.464 0.643326
## Champlain_Point.l2 -1.425e-01 5.952e-01 -0.239 0.811020
## S_S_Marie.l3 -2.208e-01 1.070e+00 -0.206 0.836719
## Detroit.l3 3.286e-01 1.208e-01 2.721 0.006961 **
## Port_Huron.l3 3.017e-02 4.286e-01 0.070 0.943931
## B_Niagara.l3 -9.127e-01 1.909e-01 -4.780 2.96e-06 ***
## Alexandria.l3 -2.104e+00 9.253e-01 -2.274 0.023795 *
## Ogdensburg.l3 1.774e+00 2.141e+00 0.828 0.408211
## Massena.l3 9.686e-01 8.139e-01 1.190 0.235127
## Champlain_Point.l3 1.070e+00 6.318e-01 1.693 0.091608 .
## S_S_Marie.l4 -7.080e-02 1.052e+00 -0.067 0.946402
## Detroit.l4 2.723e-01 1.211e-01 2.249 0.025367 *
## Port_Huron.l4 -1.841e-01 4.266e-01 -0.432 0.666461
## B_Niagara.l4 -6.221e-01 1.910e-01 -3.258 0.001275 **
## Alexandria.l4 -2.094e+00 9.339e-01 -2.242 0.025796 *
## Ogdensburg.l4 3.473e+00 2.132e+00 1.629 0.104570
## Massena.l4 1.106e+00 8.077e-01 1.370 0.171974
## Champlain_Point.l4 4.397e-01 6.438e-01 0.683 0.495236
## S_S_Marie.l5 -1.291e+00 9.905e-01 -1.304 0.193464
## Detroit.l5 5.182e-01 1.140e-01 4.544 8.53e-06 ***
## Port_Huron.l5 4.629e-01 3.991e-01 1.160 0.247141
## B_Niagara.l5 -4.732e-01 1.701e-01 -2.782 0.005808 **
## Alexandria.l5 -9.534e-01 8.539e-01 -1.116 0.265284
## Ogdensburg.l5 -9.197e-01 1.934e+00 -0.476 0.634785
## Massena.l5 7.553e-01 7.268e-01 1.039 0.299742
## Champlain_Point.l5 7.288e-01 6.160e-01 1.183 0.237822
## S_S_Marie.l6 -4.508e-01 8.742e-01 -0.516 0.606518
## Detroit.l6 2.039e-01 1.065e-01 1.915 0.056587 .
## Port_Huron.l6 -1.648e-01 3.403e-01 -0.484 0.628590
## B_Niagara.l6 -2.336e-01 1.388e-01 -1.684 0.093416 .
## Alexandria.l6 -1.516e+00 6.932e-01 -2.187 0.029663 *
## Ogdensburg.l6 1.994e+00 1.598e+00 1.248 0.213231
## Massena.l6 5.727e-01 6.198e-01 0.924 0.356322
## Champlain_Point.l6 2.918e-01 5.161e-01 0.565 0.572324
## sd1 4.078e+05 3.651e+04 11.170 < 2e-16 ***
## sd2 4.334e+03 4.534e+04 0.096 0.923913
## sd3 4.110e+04 4.046e+04 1.016 0.310609
## sd4 7.051e+04 4.773e+04 1.477 0.140864
## sd5 3.353e+05 4.032e+04 8.317 5.46e-15 ***
## sd6 6.499e+04 4.737e+04 1.372 0.171242
## sd7 1.964e+05 4.550e+04 4.317 2.26e-05 ***
## sd8 1.289e+05 3.896e+04 3.309 0.001072 **
## sd9 6.128e+04 4.247e+04 1.443 0.150250
## sd10 1.731e+05 5.088e+04 3.402 0.000778 ***
## sd11 1.730e+05 4.895e+04 3.533 0.000487 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 73460 on 255 degrees of freedom
## Multiple R-Squared: 0.9023, Adjusted R-squared: 0.8797
## F-statistic: 39.93 on 59 and 255 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation Alexandria:
## ===========================================
## Alexandria = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + S_S_Marie.l3 + Detroit.l3 + Port_Huron.l3 + B_Niagara.l3 + Alexandria.l3 + Ogdensburg.l3 + Massena.l3 + Champlain_Point.l3 + S_S_Marie.l4 + Detroit.l4 + Port_Huron.l4 + B_Niagara.l4 + Alexandria.l4 + Ogdensburg.l4 + Massena.l4 + Champlain_Point.l4 + S_S_Marie.l5 + Detroit.l5 + Port_Huron.l5 + B_Niagara.l5 + Alexandria.l5 + Ogdensburg.l5 + Massena.l5 + Champlain_Point.l5 + S_S_Marie.l6 + Detroit.l6 + Port_Huron.l6 + B_Niagara.l6 + Alexandria.l6 + Ogdensburg.l6 + Massena.l6 + Champlain_Point.l6 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11
##
## Estimate Std. Error t value Pr(>|t|)
## S_S_Marie.l1 -7.252e-02 1.327e-01 -0.546 0.585242
## Detroit.l1 3.945e-02 1.489e-02 2.649 0.008572 **
## Port_Huron.l1 1.819e-02 4.921e-02 0.370 0.712036
## B_Niagara.l1 2.843e-03 2.065e-02 0.138 0.890604
## Alexandria.l1 -8.790e-01 1.016e-01 -8.653 5.76e-16 ***
## Ogdensburg.l1 4.109e-01 2.320e-01 1.772 0.077672 .
## Massena.l1 1.213e-01 9.337e-02 1.300 0.194885
## Champlain_Point.l1 -9.833e-02 7.366e-02 -1.335 0.183082
## S_S_Marie.l2 -2.174e-01 1.503e-01 -1.446 0.149401
## Detroit.l2 4.793e-02 1.691e-02 2.835 0.004955 **
## Port_Huron.l2 5.851e-02 5.762e-02 1.015 0.310842
## B_Niagara.l2 -1.479e-02 2.493e-02 -0.593 0.553570
## Alexandria.l2 -8.070e-01 1.246e-01 -6.476 4.82e-10 ***
## Ogdensburg.l2 2.240e-01 2.844e-01 0.788 0.431530
## Massena.l2 8.963e-02 1.087e-01 0.825 0.410387
## Champlain_Point.l2 -1.000e-01 8.770e-02 -1.140 0.255166
## S_S_Marie.l3 -2.165e-01 1.577e-01 -1.373 0.171113
## Detroit.l3 4.365e-02 1.779e-02 2.453 0.014829 *
## Port_Huron.l3 7.665e-02 6.315e-02 1.214 0.225913
## B_Niagara.l3 -5.647e-02 2.813e-02 -2.007 0.045790 *
## Alexandria.l3 -7.991e-01 1.363e-01 -5.862 1.41e-08 ***
## Ogdensburg.l3 4.115e-01 3.155e-01 1.304 0.193329
## Massena.l3 1.395e-01 1.199e-01 1.163 0.245784
## Champlain_Point.l3 1.029e-01 9.309e-02 1.106 0.269836
## S_S_Marie.l4 -1.825e-01 1.550e-01 -1.178 0.240091
## Detroit.l4 3.479e-02 1.784e-02 1.950 0.052223 .
## Port_Huron.l4 5.278e-02 6.286e-02 0.840 0.401938
## B_Niagara.l4 -4.338e-02 2.814e-02 -1.542 0.124349
## Alexandria.l4 -5.651e-01 1.376e-01 -4.107 5.41e-05 ***
## Ogdensburg.l4 3.785e-01 3.141e-01 1.205 0.229309
## Massena.l4 2.998e-02 1.190e-01 0.252 0.801344
## Champlain_Point.l4 7.224e-02 9.486e-02 0.762 0.447046
## S_S_Marie.l5 -2.276e-01 1.459e-01 -1.560 0.120043
## Detroit.l5 4.577e-02 1.680e-02 2.724 0.006897 **
## Port_Huron.l5 1.781e-01 5.880e-02 3.029 0.002702 **
## B_Niagara.l5 -3.989e-02 2.507e-02 -1.591 0.112752
## Alexandria.l5 -3.535e-01 1.258e-01 -2.810 0.005344 **
## Ogdensburg.l5 3.971e-02 2.849e-01 0.139 0.889261
## Massena.l5 5.285e-02 1.071e-01 0.494 0.622062
## Champlain_Point.l5 5.270e-02 9.076e-02 0.581 0.561989
## S_S_Marie.l6 -1.692e-01 1.288e-01 -1.314 0.190155
## Detroit.l6 2.319e-02 1.569e-02 1.478 0.140519
## Port_Huron.l6 3.197e-02 5.014e-02 0.638 0.524335
## B_Niagara.l6 -2.169e-02 2.044e-02 -1.061 0.289668
## Alexandria.l6 -3.073e-01 1.021e-01 -3.009 0.002888 **
## Ogdensburg.l6 4.322e-01 2.354e-01 1.836 0.067543 .
## Massena.l6 4.305e-02 9.132e-02 0.471 0.637701
## Champlain_Point.l6 1.635e-02 7.605e-02 0.215 0.829961
## sd1 5.927e+04 5.379e+03 11.019 < 2e-16 ***
## sd2 2.943e+03 6.680e+03 0.441 0.659902
## sd3 6.459e+03 5.961e+03 1.084 0.279578
## sd4 1.059e+04 7.033e+03 1.506 0.133331
## sd5 5.290e+04 5.941e+03 8.905 < 2e-16 ***
## sd6 1.266e+04 6.979e+03 1.814 0.070834 .
## sd7 2.894e+04 6.703e+03 4.317 2.26e-05 ***
## sd8 1.594e+04 5.741e+03 2.777 0.005897 **
## sd9 1.325e+04 6.257e+03 2.117 0.035191 *
## sd10 2.436e+04 7.497e+03 3.249 0.001315 **
## sd11 2.560e+04 7.212e+03 3.550 0.000459 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 10820 on 255 degrees of freedom
## Multiple R-Squared: 0.9045, Adjusted R-squared: 0.8824
## F-statistic: 40.94 on 59 and 255 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation Ogdensburg:
## ===========================================
## Ogdensburg = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + S_S_Marie.l3 + Detroit.l3 + Port_Huron.l3 + B_Niagara.l3 + Alexandria.l3 + Ogdensburg.l3 + Massena.l3 + Champlain_Point.l3 + S_S_Marie.l4 + Detroit.l4 + Port_Huron.l4 + B_Niagara.l4 + Alexandria.l4 + Ogdensburg.l4 + Massena.l4 + Champlain_Point.l4 + S_S_Marie.l5 + Detroit.l5 + Port_Huron.l5 + B_Niagara.l5 + Alexandria.l5 + Ogdensburg.l5 + Massena.l5 + Champlain_Point.l5 + S_S_Marie.l6 + Detroit.l6 + Port_Huron.l6 + B_Niagara.l6 + Alexandria.l6 + Ogdensburg.l6 + Massena.l6 + Champlain_Point.l6 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11
##
## Estimate Std. Error t value Pr(>|t|)
## S_S_Marie.l1 -3.271e-02 5.083e-02 -0.643 0.52052
## Detroit.l1 3.804e-03 5.703e-03 0.667 0.50538
## Port_Huron.l1 2.680e-02 1.885e-02 1.422 0.15634
## B_Niagara.l1 3.670e-03 7.909e-03 0.464 0.64302
## Alexandria.l1 -5.414e-02 3.891e-02 -1.391 0.16533
## Ogdensburg.l1 -6.701e-01 8.884e-02 -7.542 8.14e-13 ***
## Massena.l1 -6.729e-03 3.576e-02 -0.188 0.85089
## Champlain_Point.l1 6.412e-03 2.821e-02 0.227 0.82040
## S_S_Marie.l2 1.442e-02 5.758e-02 0.250 0.80242
## Detroit.l2 7.999e-03 6.477e-03 1.235 0.21796
## Port_Huron.l2 3.122e-02 2.207e-02 1.415 0.15835
## B_Niagara.l2 2.108e-03 9.549e-03 0.221 0.82547
## Alexandria.l2 -6.027e-02 4.773e-02 -1.263 0.20781
## Ogdensburg.l2 -6.615e-01 1.089e-01 -6.073 4.54e-09 ***
## Massena.l2 1.066e-02 4.163e-02 0.256 0.79812
## Champlain_Point.l2 -6.816e-02 3.359e-02 -2.029 0.04349 *
## S_S_Marie.l3 3.301e-02 6.041e-02 0.546 0.58526
## Detroit.l3 1.276e-02 6.815e-03 1.872 0.06236 .
## Port_Huron.l3 4.381e-02 2.419e-02 1.811 0.07126 .
## B_Niagara.l3 -1.018e-02 1.078e-02 -0.945 0.34549
## Alexandria.l3 -8.770e-02 5.222e-02 -1.679 0.09430 .
## Ogdensburg.l3 -6.688e-01 1.208e-01 -5.535 7.72e-08 ***
## Massena.l3 -6.504e-03 4.593e-02 -0.142 0.88751
## Champlain_Point.l3 1.230e-02 3.566e-02 0.345 0.73033
## S_S_Marie.l4 -6.064e-04 5.938e-02 -0.010 0.99186
## Detroit.l4 1.007e-02 6.832e-03 1.475 0.14156
## Port_Huron.l4 1.131e-02 2.408e-02 0.470 0.63889
## B_Niagara.l4 -2.552e-03 1.078e-02 -0.237 0.81297
## Alexandria.l4 -8.894e-02 5.271e-02 -1.687 0.09274 .
## Ogdensburg.l4 -3.533e-01 1.203e-01 -2.936 0.00363 **
## Massena.l4 -2.568e-03 4.559e-02 -0.056 0.95512
## Champlain_Point.l4 1.452e-02 3.633e-02 0.400 0.68973
## S_S_Marie.l5 -5.238e-02 5.590e-02 -0.937 0.34958
## Detroit.l5 1.748e-02 6.436e-03 2.716 0.00706 **
## Port_Huron.l5 1.870e-03 2.252e-02 0.083 0.93390
## B_Niagara.l5 4.916e-03 9.600e-03 0.512 0.60902
## Alexandria.l5 -7.450e-02 4.819e-02 -1.546 0.12336
## Ogdensburg.l5 -2.877e-01 1.091e-01 -2.636 0.00890 **
## Massena.l5 6.123e-04 4.102e-02 0.015 0.98810
## Champlain_Point.l5 4.308e-02 3.476e-02 1.239 0.21634
## S_S_Marie.l6 -9.921e-03 4.933e-02 -0.201 0.84079
## Detroit.l6 6.199e-03 6.009e-03 1.032 0.30325
## Port_Huron.l6 -1.003e-02 1.920e-02 -0.522 0.60185
## B_Niagara.l6 -2.757e-03 7.830e-03 -0.352 0.72506
## Alexandria.l6 -2.460e-02 3.912e-02 -0.629 0.53002
## Ogdensburg.l6 -1.378e-01 9.017e-02 -1.528 0.12769
## Massena.l6 3.611e-02 3.498e-02 1.033 0.30281
## Champlain_Point.l6 1.200e-02 2.913e-02 0.412 0.68075
## sd1 1.162e+04 2.060e+03 5.640 4.51e-08 ***
## sd2 -1.744e+03 2.559e+03 -0.682 0.49617
## sd3 1.270e+03 2.283e+03 0.556 0.57858
## sd4 -4.923e+02 2.694e+03 -0.183 0.85512
## sd5 1.091e+04 2.275e+03 4.797 2.75e-06 ***
## sd6 3.927e+02 2.673e+03 0.147 0.88333
## sd7 6.471e+03 2.568e+03 2.520 0.01233 *
## sd8 4.406e+03 2.199e+03 2.004 0.04613 *
## sd9 -3.331e+01 2.397e+03 -0.014 0.98892
## sd10 3.689e+03 2.872e+03 1.285 0.20006
## sd11 5.387e+03 2.762e+03 1.950 0.05225 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 4146 on 255 degrees of freedom
## Multiple R-Squared: 0.7873, Adjusted R-squared: 0.738
## F-statistic: 15.99 on 59 and 255 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation Massena:
## ========================================
## Massena = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + S_S_Marie.l3 + Detroit.l3 + Port_Huron.l3 + B_Niagara.l3 + Alexandria.l3 + Ogdensburg.l3 + Massena.l3 + Champlain_Point.l3 + S_S_Marie.l4 + Detroit.l4 + Port_Huron.l4 + B_Niagara.l4 + Alexandria.l4 + Ogdensburg.l4 + Massena.l4 + Champlain_Point.l4 + S_S_Marie.l5 + Detroit.l5 + Port_Huron.l5 + B_Niagara.l5 + Alexandria.l5 + Ogdensburg.l5 + Massena.l5 + Champlain_Point.l5 + S_S_Marie.l6 + Detroit.l6 + Port_Huron.l6 + B_Niagara.l6 + Alexandria.l6 + Ogdensburg.l6 + Massena.l6 + Champlain_Point.l6 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11
##
## Estimate Std. Error t value Pr(>|t|)
## S_S_Marie.l1 5.905e-03 1.044e-01 0.057 0.954961
## Detroit.l1 5.720e-03 1.172e-02 0.488 0.625901
## Port_Huron.l1 1.850e-02 3.873e-02 0.478 0.633326
## B_Niagara.l1 2.813e-02 1.625e-02 1.731 0.084630 .
## Alexandria.l1 -1.152e-01 7.995e-02 -1.441 0.150911
## Ogdensburg.l1 1.821e-01 1.826e-01 0.998 0.319428
## Massena.l1 -5.562e-01 7.348e-02 -7.569 6.86e-13 ***
## Champlain_Point.l1 -1.241e-01 5.797e-02 -2.141 0.033262 *
## S_S_Marie.l2 -5.001e-02 1.183e-01 -0.423 0.672842
## Detroit.l2 2.957e-02 1.331e-02 2.222 0.027146 *
## Port_Huron.l2 4.633e-02 4.535e-02 1.022 0.307861
## B_Niagara.l2 -1.492e-02 1.962e-02 -0.761 0.447611
## Alexandria.l2 4.154e-02 9.807e-02 0.424 0.672241
## Ogdensburg.l2 8.696e-02 2.238e-01 0.389 0.697934
## Massena.l2 -5.968e-01 8.555e-02 -6.976 2.60e-11 ***
## Champlain_Point.l2 -1.191e-01 6.902e-02 -1.725 0.085748 .
## S_S_Marie.l3 -9.438e-02 1.241e-01 -0.760 0.447752
## Detroit.l3 1.555e-02 1.400e-02 1.110 0.267960
## Port_Huron.l3 -2.756e-02 4.970e-02 -0.555 0.579706
## B_Niagara.l3 -1.307e-02 2.214e-02 -0.590 0.555438
## Alexandria.l3 -3.065e-02 1.073e-01 -0.286 0.775370
## Ogdensburg.l3 6.297e-02 2.483e-01 0.254 0.800016
## Massena.l3 -4.092e-01 9.438e-02 -4.336 2.09e-05 ***
## Champlain_Point.l3 6.431e-02 7.326e-02 0.878 0.380861
## S_S_Marie.l4 -8.992e-02 1.220e-01 -0.737 0.461761
## Detroit.l4 1.239e-02 1.404e-02 0.883 0.378238
## Port_Huron.l4 -3.725e-02 4.947e-02 -0.753 0.452161
## B_Niagara.l4 -8.908e-04 2.214e-02 -0.040 0.967941
## Alexandria.l4 -7.376e-02 1.083e-01 -0.681 0.496418
## Ogdensburg.l4 1.882e-01 2.472e-01 0.761 0.447155
## Massena.l4 -2.916e-01 9.367e-02 -3.113 0.002064 **
## Champlain_Point.l4 -1.190e-03 7.465e-02 -0.016 0.987300
## S_S_Marie.l5 5.712e-02 1.149e-01 0.497 0.619374
## Detroit.l5 2.991e-02 1.322e-02 2.262 0.024533 *
## Port_Huron.l5 5.023e-02 4.628e-02 1.085 0.278744
## B_Niagara.l5 -1.080e-02 1.973e-02 -0.547 0.584573
## Alexandria.l5 -8.068e-02 9.902e-02 -0.815 0.415952
## Ogdensburg.l5 -2.299e-01 2.242e-01 -1.025 0.306166
## Massena.l5 -2.018e-01 8.428e-02 -2.394 0.017382 *
## Champlain_Point.l5 -2.061e-02 7.143e-02 -0.289 0.773143
## S_S_Marie.l6 5.377e-02 1.014e-01 0.530 0.596285
## Detroit.l6 -1.524e-02 1.235e-02 -1.234 0.218188
## Port_Huron.l6 5.313e-02 3.946e-02 1.346 0.179358
## B_Niagara.l6 -3.771e-03 1.609e-02 -0.234 0.814876
## Alexandria.l6 -1.217e-01 8.038e-02 -1.514 0.131389
## Ogdensburg.l6 1.708e-01 1.853e-01 0.922 0.357573
## Massena.l6 -8.831e-02 7.187e-02 -1.229 0.220281
## Champlain_Point.l6 -5.804e-02 5.985e-02 -0.970 0.333053
## sd1 2.353e+04 4.233e+03 5.558 6.86e-08 ***
## sd2 2.986e+03 5.257e+03 0.568 0.570586
## sd3 4.672e+03 4.691e+03 0.996 0.320228
## sd4 3.798e+03 5.535e+03 0.686 0.493207
## sd5 2.531e+04 4.675e+03 5.413 1.43e-07 ***
## sd6 8.523e+03 5.493e+03 1.552 0.121952
## sd7 1.909e+04 5.276e+03 3.618 0.000357 ***
## sd8 1.095e+04 4.518e+03 2.424 0.016031 *
## sd9 8.379e+03 4.924e+03 1.702 0.090061 .
## sd10 2.394e+04 5.900e+03 4.057 6.61e-05 ***
## sd11 9.089e+03 5.676e+03 1.601 0.110581
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 8519 on 255 degrees of freedom
## Multiple R-Squared: 0.7254, Adjusted R-squared: 0.6619
## F-statistic: 11.42 on 59 and 255 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation Champlain_Point:
## ================================================
## Champlain_Point = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + S_S_Marie.l3 + Detroit.l3 + Port_Huron.l3 + B_Niagara.l3 + Alexandria.l3 + Ogdensburg.l3 + Massena.l3 + Champlain_Point.l3 + S_S_Marie.l4 + Detroit.l4 + Port_Huron.l4 + B_Niagara.l4 + Alexandria.l4 + Ogdensburg.l4 + Massena.l4 + Champlain_Point.l4 + S_S_Marie.l5 + Detroit.l5 + Port_Huron.l5 + B_Niagara.l5 + Alexandria.l5 + Ogdensburg.l5 + Massena.l5 + Champlain_Point.l5 + S_S_Marie.l6 + Detroit.l6 + Port_Huron.l6 + B_Niagara.l6 + Alexandria.l6 + Ogdensburg.l6 + Massena.l6 + Champlain_Point.l6 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11
##
## Estimate Std. Error t value Pr(>|t|)
## S_S_Marie.l1 -3.730e-02 1.914e-01 -0.195 0.845616
## Detroit.l1 5.040e-02 2.147e-02 2.347 0.019684 *
## Port_Huron.l1 4.802e-02 7.097e-02 0.677 0.499284
## B_Niagara.l1 -1.759e-02 2.978e-02 -0.591 0.555158
## Alexandria.l1 -7.324e-02 1.465e-01 -0.500 0.617516
## Ogdensburg.l1 1.566e-02 3.345e-01 0.047 0.962698
## Massena.l1 1.462e-01 1.346e-01 1.086 0.278700
## Champlain_Point.l1 -7.432e-01 1.062e-01 -6.997 2.30e-11 ***
## S_S_Marie.l2 -1.413e-01 2.168e-01 -0.652 0.515041
## Detroit.l2 5.962e-02 2.438e-02 2.445 0.015163 *
## Port_Huron.l2 1.129e-01 8.309e-02 1.359 0.175419
## B_Niagara.l2 -4.556e-02 3.595e-02 -1.267 0.206192
## Alexandria.l2 -2.157e-01 1.797e-01 -1.201 0.231033
## Ogdensburg.l2 -1.937e-01 4.101e-01 -0.472 0.637158
## Massena.l2 6.260e-02 1.567e-01 0.399 0.689944
## Champlain_Point.l2 -7.003e-01 1.265e-01 -5.538 7.62e-08 ***
## S_S_Marie.l3 -5.393e-02 2.274e-01 -0.237 0.812771
## Detroit.l3 8.323e-02 2.566e-02 3.244 0.001337 **
## Port_Huron.l3 8.437e-02 9.106e-02 0.927 0.355038
## B_Niagara.l3 -8.168e-02 4.057e-02 -2.013 0.045132 *
## Alexandria.l3 -3.590e-01 1.966e-01 -1.826 0.069042 .
## Ogdensburg.l3 -1.789e-01 4.549e-01 -0.393 0.694423
## Massena.l3 1.220e-01 1.729e-01 0.705 0.481249
## Champlain_Point.l3 -3.641e-01 1.342e-01 -2.713 0.007132 **
## S_S_Marie.l4 1.204e-01 2.235e-01 0.538 0.590741
## Detroit.l4 5.815e-02 2.572e-02 2.261 0.024628 *
## Port_Huron.l4 9.490e-02 9.065e-02 1.047 0.296155
## B_Niagara.l4 -8.667e-02 4.057e-02 -2.136 0.033614 *
## Alexandria.l4 -2.374e-01 1.984e-01 -1.196 0.232741
## Ogdensburg.l4 8.503e-03 4.530e-01 0.019 0.985039
## Massena.l4 8.573e-02 1.716e-01 0.500 0.617854
## Champlain_Point.l4 -3.202e-01 1.368e-01 -2.341 0.020026 *
## S_S_Marie.l5 -1.628e-02 2.105e-01 -0.077 0.938398
## Detroit.l5 9.495e-02 2.423e-02 3.919 0.000114 ***
## Port_Huron.l5 1.256e-01 8.480e-02 1.481 0.139796
## B_Niagara.l5 -7.391e-02 3.614e-02 -2.045 0.041908 *
## Alexandria.l5 -1.427e-01 1.814e-01 -0.786 0.432377
## Ogdensburg.l5 3.745e-02 4.109e-01 0.091 0.927454
## Massena.l5 1.064e-01 1.544e-01 0.689 0.491306
## Champlain_Point.l5 -1.673e-01 1.309e-01 -1.278 0.202330
## S_S_Marie.l6 -1.817e-01 1.857e-01 -0.978 0.328860
## Detroit.l6 5.712e-02 2.262e-02 2.525 0.012182 *
## Port_Huron.l6 -3.439e-02 7.230e-02 -0.476 0.634717
## B_Niagara.l6 -3.296e-02 2.948e-02 -1.118 0.264645
## Alexandria.l6 -8.702e-02 1.473e-01 -0.591 0.555144
## Ogdensburg.l6 4.505e-02 3.395e-01 0.133 0.894518
## Massena.l6 2.499e-02 1.317e-01 0.190 0.849643
## Champlain_Point.l6 -5.048e-02 1.097e-01 -0.460 0.645687
## sd1 8.427e+04 7.757e+03 10.865 < 2e-16 ***
## sd2 -1.094e+03 9.633e+03 -0.114 0.909666
## sd3 1.237e+04 8.596e+03 1.439 0.151409
## sd4 1.446e+04 1.014e+04 1.426 0.155230
## sd5 7.954e+04 8.567e+03 9.284 < 2e-16 ***
## sd6 1.572e+04 1.006e+04 1.562 0.119627
## sd7 3.272e+04 9.667e+03 3.385 0.000823 ***
## sd8 2.605e+04 8.279e+03 3.147 0.001846 **
## sd9 9.115e+03 9.023e+03 1.010 0.313333
## sd10 3.352e+04 1.081e+04 3.100 0.002150 **
## sd11 3.167e+04 1.040e+04 3.045 0.002572 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 15610 on 255 degrees of freedom
## Multiple R-Squared: 0.902, Adjusted R-squared: 0.8793
## F-statistic: 39.78 on 59 and 255 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## S_S_Marie Detroit Port_Huron B_Niagara Alexandria Ogdensburg
## S_S_Marie 73242283 2.775e+08 1.372e+08 4.949e+08 60274661 22184127
## Detroit 277475716 3.628e+09 9.115e+08 2.707e+09 318203627 106120836
## Port_Huron 137197183 9.115e+08 5.765e+08 1.368e+09 172115518 58829189
## B_Niagara 494889515 2.707e+09 1.368e+09 5.382e+09 621768243 186936601
## Alexandria 60274661 3.182e+08 1.721e+08 6.218e+08 117040519 25738662
## Ogdensburg 22184127 1.061e+08 5.883e+07 1.869e+08 25738662 17179219
## Massena 32633829 2.209e+08 9.801e+07 3.012e+08 40893775 10370716
## Champlain_Point 88352541 4.718e+08 2.682e+08 8.688e+08 114418973 43178371
## Massena Champlain_Point
## S_S_Marie 32633829 88352541
## Detroit 220904248 471827833
## Port_Huron 98007913 268178013
## B_Niagara 301203218 868764447
## Alexandria 40893775 114418973
## Ogdensburg 10370716 43178371
## Massena 72478594 58835239
## Champlain_Point 58835239 243642990
##
## Correlation matrix of residuals:
## S_S_Marie Detroit Port_Huron B_Niagara Alexandria Ogdensburg
## S_S_Marie 1.0000 0.5383 0.6677 0.7883 0.6510 0.6254
## Detroit 0.5383 1.0000 0.6302 0.6125 0.4883 0.4251
## Port_Huron 0.6677 0.6302 1.0000 0.7768 0.6626 0.5911
## B_Niagara 0.7883 0.6125 0.7768 1.0000 0.7834 0.6148
## Alexandria 0.6510 0.4883 0.6626 0.7834 1.0000 0.5740
## Ogdensburg 0.6254 0.4251 0.5911 0.6148 0.5740 1.0000
## Massena 0.4479 0.4308 0.4795 0.4823 0.4440 0.2939
## Champlain_Point 0.6614 0.5018 0.7156 0.7587 0.6776 0.6674
## Massena Champlain_Point
## S_S_Marie 0.4479 0.6614
## Detroit 0.4308 0.5018
## Port_Huron 0.4795 0.7156
## B_Niagara 0.4823 0.7587
## Alexandria 0.4440 0.6776
## Ogdensburg 0.2939 0.6674
## Massena 1.0000 0.4427
## Champlain_Point 0.4427 1.0000
selected_order <- var.a$selection$ic[1, "lag"]
# Print the selected lag order
print(selected_order)
## NULL
VARselect(stnry_ts, lag.max=8, type="none", season = 12)$selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 6 3 1 6
var <- vars::VAR(stnry_ts, p=2, season = 12, type = "none", ic = c("SC"))
summary(var)
##
## VAR Estimation Results:
## =========================
## Endogenous variables: S_S_Marie, Detroit, Port_Huron, B_Niagara, Alexandria, Ogdensburg, Massena, Champlain_Point
## Deterministic variables: none
## Sample size: 318
## Log Likelihood: -27538.11
## Roots of the characteristic polynomial:
## 0.7037 0.7037 0.6716 0.6716 0.6145 0.6145 0.5986 0.5986 0.5798 0.5798 0.4839 0.4839 0.4462 0.4462 0.3807 0.3807
## Call:
## vars::VAR(y = stnry_ts, p = 2, type = "none", season = 12L, ic = c("SC"))
##
##
## Estimation results for equation S_S_Marie:
## ==========================================
## S_S_Marie = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11
##
## Estimate Std. Error t value Pr(>|t|)
## S_S_Marie.l1 -5.197e-01 1.045e-01 -4.974 1.13e-06 ***
## Detroit.l1 2.014e-02 1.229e-02 1.638 0.1024
## Port_Huron.l1 -1.690e-02 3.867e-02 -0.437 0.6624
## B_Niagara.l1 -7.944e-03 1.464e-02 -0.543 0.5877
## Alexandria.l1 4.776e-02 7.806e-02 0.612 0.5412
## Ogdensburg.l1 2.901e-01 1.727e-01 1.680 0.0941 .
## Massena.l1 7.303e-02 7.050e-02 1.036 0.3011
## Champlain_Point.l1 -1.111e-02 5.720e-02 -0.194 0.8462
## S_S_Marie.l2 -2.348e-01 1.035e-01 -2.268 0.0241 *
## Detroit.l2 -5.614e-03 1.237e-02 -0.454 0.6502
## Port_Huron.l2 7.187e-02 3.818e-02 1.883 0.0608 .
## B_Niagara.l2 -1.413e-02 1.490e-02 -0.948 0.3437
## Alexandria.l2 3.433e-02 7.915e-02 0.434 0.6648
## Ogdensburg.l2 7.881e-03 1.701e-01 0.046 0.9631
## Massena.l2 -1.208e-02 6.877e-02 -0.176 0.8606
## Champlain_Point.l2 -9.436e-02 5.783e-02 -1.632 0.1038
## sd1 3.280e+04 3.278e+03 10.006 < 2e-16 ***
## sd2 -1.550e+04 3.790e+03 -4.090 5.59e-05 ***
## sd3 6.309e+03 3.900e+03 1.618 0.1068
## sd4 1.069e+03 4.258e+03 0.251 0.8019
## sd5 3.947e+04 3.105e+03 12.711 < 2e-16 ***
## sd6 9.188e+03 3.801e+03 2.417 0.0163 *
## sd7 8.780e+03 3.898e+03 2.253 0.0250 *
## sd8 6.256e+03 3.996e+03 1.565 0.1186
## sd9 -1.481e+03 3.320e+03 -0.446 0.6558
## sd10 2.138e+04 3.283e+03 6.510 3.29e-10 ***
## sd11 1.863e+04 3.171e+03 5.873 1.17e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 9929 on 291 degrees of freedom
## Multiple R-Squared: 0.7805, Adjusted R-squared: 0.7601
## F-statistic: 38.32 on 27 and 291 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation Detroit:
## ========================================
## Detroit = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11
##
## Estimate Std. Error t value Pr(>|t|)
## S_S_Marie.l1 6.280e-01 6.768e-01 0.928 0.35417
## Detroit.l1 -3.595e-01 7.962e-02 -4.515 9.21e-06 ***
## Port_Huron.l1 4.796e-02 2.505e-01 0.191 0.84830
## B_Niagara.l1 -1.173e-01 9.481e-02 -1.238 0.21684
## Alexandria.l1 2.860e-02 5.057e-01 0.057 0.95493
## Ogdensburg.l1 3.400e-01 1.119e+00 0.304 0.76137
## Massena.l1 1.910e-01 4.567e-01 0.418 0.67609
## Champlain_Point.l1 1.490e-01 3.705e-01 0.402 0.68790
## S_S_Marie.l2 2.043e-01 6.706e-01 0.305 0.76083
## Detroit.l2 -2.220e-01 8.010e-02 -2.772 0.00593 **
## Port_Huron.l2 1.622e-01 2.473e-01 0.656 0.51229
## B_Niagara.l2 -6.244e-02 9.654e-02 -0.647 0.51825
## Alexandria.l2 1.237e-02 5.127e-01 0.024 0.98076
## Ogdensburg.l2 -8.984e-01 1.102e+00 -0.815 0.41549
## Massena.l2 1.047e-01 4.454e-01 0.235 0.81429
## Champlain_Point.l2 -1.072e-01 3.745e-01 -0.286 0.77488
## sd1 9.517e+04 2.123e+04 4.482 1.06e-05 ***
## sd2 -7.417e+04 2.455e+04 -3.021 0.00274 **
## sd3 3.333e+04 2.526e+04 1.319 0.18810
## sd4 2.481e+04 2.758e+04 0.900 0.36896
## sd5 6.394e+04 2.011e+04 3.179 0.00164 **
## sd6 7.328e+04 2.462e+04 2.977 0.00316 **
## sd7 6.247e+04 2.525e+04 2.474 0.01392 *
## sd8 2.044e+04 2.589e+04 0.790 0.43046
## sd9 -6.945e+03 2.151e+04 -0.323 0.74695
## sd10 6.843e+04 2.127e+04 3.217 0.00144 **
## sd11 1.184e+04 2.054e+04 0.576 0.56481
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 64320 on 291 degrees of freedom
## Multiple R-Squared: 0.4226, Adjusted R-squared: 0.369
## F-statistic: 7.887 on 27 and 291 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation Port_Huron:
## ===========================================
## Port_Huron = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11
##
## Estimate Std. Error t value Pr(>|t|)
## S_S_Marie.l1 -8.881e-02 2.788e-01 -0.319 0.7503
## Detroit.l1 3.670e-02 3.280e-02 1.119 0.2641
## Port_Huron.l1 -4.995e-01 1.032e-01 -4.841 2.10e-06 ***
## B_Niagara.l1 1.081e-02 3.905e-02 0.277 0.7820
## Alexandria.l1 3.787e-02 2.083e-01 0.182 0.8558
## Ogdensburg.l1 4.800e-01 4.607e-01 1.042 0.2983
## Massena.l1 1.857e-01 1.881e-01 0.988 0.3242
## Champlain_Point.l1 -1.498e-01 1.526e-01 -0.981 0.3272
## S_S_Marie.l2 2.562e-01 2.762e-01 0.927 0.3545
## Detroit.l2 2.565e-03 3.299e-02 0.078 0.9381
## Port_Huron.l2 -1.750e-01 1.018e-01 -1.718 0.0868 .
## B_Niagara.l2 -3.538e-02 3.976e-02 -0.890 0.3743
## Alexandria.l2 -4.062e-02 2.112e-01 -0.192 0.8476
## Ogdensburg.l2 -1.040e-02 4.538e-01 -0.023 0.9817
## Massena.l2 6.288e-02 1.835e-01 0.343 0.7320
## Champlain_Point.l2 -2.767e-01 1.543e-01 -1.793 0.0739 .
## sd1 7.782e+04 8.745e+03 8.898 < 2e-16 ***
## sd2 -5.151e+04 1.011e+04 -5.095 6.28e-07 ***
## sd3 -4.840e+03 1.041e+04 -0.465 0.6422
## sd4 -1.759e+04 1.136e+04 -1.549 0.1226
## sd5 9.130e+04 8.284e+03 11.021 < 2e-16 ***
## sd6 4.232e+04 1.014e+04 4.173 3.97e-05 ***
## sd7 2.027e+04 1.040e+04 1.949 0.0522 .
## sd8 1.262e+04 1.066e+04 1.184 0.2374
## sd9 -7.441e+03 8.858e+03 -0.840 0.4016
## sd10 4.042e+04 8.760e+03 4.614 5.94e-06 ***
## sd11 1.811e+04 8.461e+03 2.141 0.0331 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 26490 on 291 degrees of freedom
## Multiple R-Squared: 0.7626, Adjusted R-squared: 0.7405
## F-statistic: 34.61 on 27 and 291 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation B_Niagara:
## ==========================================
## B_Niagara = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11
##
## Estimate Std. Error t value Pr(>|t|)
## S_S_Marie.l1 -4.521e-01 9.527e-01 -0.475 0.635480
## Detroit.l1 1.992e-01 1.121e-01 1.777 0.076568 .
## Port_Huron.l1 6.945e-02 3.526e-01 0.197 0.843992
## B_Niagara.l1 -4.970e-01 1.335e-01 -3.724 0.000235 ***
## Alexandria.l1 -5.166e-01 7.118e-01 -0.726 0.468560
## Ogdensburg.l1 1.035e+00 1.575e+00 0.657 0.511523
## Massena.l1 5.466e-01 6.428e-01 0.850 0.395807
## Champlain_Point.l1 2.322e-01 5.216e-01 0.445 0.656481
## S_S_Marie.l2 6.776e-01 9.440e-01 0.718 0.473491
## Detroit.l2 5.071e-02 1.128e-01 0.450 0.653233
## Port_Huron.l2 6.840e-01 3.481e-01 1.965 0.050351 .
## B_Niagara.l2 -4.415e-01 1.359e-01 -3.249 0.001295 **
## Alexandria.l2 -7.419e-02 7.217e-01 -0.103 0.918198
## Ogdensburg.l2 -8.016e-01 1.551e+00 -0.517 0.605666
## Massena.l2 -1.021e+00 6.270e-01 -1.628 0.104565
## Champlain_Point.l2 -4.169e-01 5.272e-01 -0.791 0.429735
## sd1 3.390e+05 2.989e+04 11.344 < 2e-16 ***
## sd2 -1.968e+05 3.455e+04 -5.696 3.00e-08 ***
## sd3 -2.074e+04 3.556e+04 -0.583 0.560279
## sd4 -7.119e+04 3.882e+04 -1.834 0.067712 .
## sd5 3.920e+05 2.831e+04 13.846 < 2e-16 ***
## sd6 7.409e+04 3.466e+04 2.138 0.033373 *
## sd7 1.556e+04 3.554e+04 0.438 0.661835
## sd8 1.921e+04 3.644e+04 0.527 0.598533
## sd9 -4.438e+04 3.027e+04 -1.466 0.143756
## sd10 1.468e+05 2.994e+04 4.904 1.57e-06 ***
## sd11 8.752e+04 2.892e+04 3.027 0.002693 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 90540 on 291 degrees of freedom
## Multiple R-Squared: 0.835, Adjusted R-squared: 0.8196
## F-statistic: 54.53 on 27 and 291 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation Alexandria:
## ===========================================
## Alexandria = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11
##
## Estimate Std. Error t value Pr(>|t|)
## S_S_Marie.l1 -5.090e-02 1.380e-01 -0.369 0.712589
## Detroit.l1 3.644e-02 1.624e-02 2.244 0.025608 *
## Port_Huron.l1 -1.067e-02 5.109e-02 -0.209 0.834677
## B_Niagara.l1 1.489e-02 1.934e-02 0.770 0.442107
## Alexandria.l1 -6.303e-01 1.031e-01 -6.111 3.17e-09 ***
## Ogdensburg.l1 2.136e-01 2.281e-01 0.936 0.349930
## Massena.l1 4.194e-02 9.315e-02 0.450 0.652869
## Champlain_Point.l1 -3.882e-02 7.558e-02 -0.514 0.607896
## S_S_Marie.l2 -6.416e-02 1.368e-01 -0.469 0.639410
## Detroit.l2 1.737e-02 1.634e-02 1.063 0.288565
## Port_Huron.l2 3.517e-02 5.044e-02 0.697 0.486116
## B_Niagara.l2 7.853e-03 1.969e-02 0.399 0.690304
## Alexandria.l2 -2.986e-01 1.046e-01 -2.856 0.004604 **
## Ogdensburg.l2 -3.167e-02 2.247e-01 -0.141 0.888030
## Massena.l2 -6.896e-02 9.086e-02 -0.759 0.448449
## Champlain_Point.l2 -1.312e-01 7.640e-02 -1.717 0.087062 .
## sd1 5.190e+04 4.331e+03 11.984 < 2e-16 ***
## sd2 -2.432e+04 5.007e+03 -4.857 1.95e-06 ***
## sd3 5.803e+02 5.153e+03 0.113 0.910410
## sd4 -8.198e+03 5.625e+03 -1.457 0.146078
## sd5 6.237e+04 4.102e+03 15.204 < 2e-16 ***
## sd6 1.398e+04 5.022e+03 2.784 0.005717 **
## sd7 1.285e+03 5.150e+03 0.250 0.803117
## sd8 1.310e+03 5.280e+03 0.248 0.804275
## sd9 3.891e+03 4.386e+03 0.887 0.375735
## sd10 2.551e+04 4.338e+03 5.881 1.12e-08 ***
## sd11 1.398e+04 4.190e+03 3.336 0.000961 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 13120 on 291 degrees of freedom
## Multiple R-Squared: 0.8435, Adjusted R-squared: 0.829
## F-statistic: 58.1 on 27 and 291 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation Ogdensburg:
## ===========================================
## Ogdensburg = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11
##
## Estimate Std. Error t value Pr(>|t|)
## S_S_Marie.l1 -6.859e-02 4.859e-02 -1.412 0.159128
## Detroit.l1 3.304e-03 5.717e-03 0.578 0.563738
## Port_Huron.l1 4.363e-03 1.798e-02 0.243 0.808457
## B_Niagara.l1 8.812e-03 6.807e-03 1.295 0.196484
## Alexandria.l1 -2.857e-02 3.630e-02 -0.787 0.431973
## Ogdensburg.l1 -4.377e-01 8.030e-02 -5.451 1.07e-07 ***
## Massena.l1 -1.858e-03 3.279e-02 -0.057 0.954844
## Champlain_Point.l1 2.008e-02 2.660e-02 0.755 0.451042
## S_S_Marie.l2 -2.813e-02 4.815e-02 -0.584 0.559557
## Detroit.l2 2.498e-04 5.751e-03 0.043 0.965378
## Port_Huron.l2 5.464e-03 1.775e-02 0.308 0.758459
## B_Niagara.l2 5.316e-03 6.931e-03 0.767 0.443712
## Alexandria.l2 5.381e-03 3.681e-02 0.146 0.883866
## Ogdensburg.l2 -2.852e-01 7.910e-02 -3.606 0.000366 ***
## Massena.l2 1.115e-02 3.198e-02 0.349 0.727605
## Champlain_Point.l2 -5.637e-02 2.689e-02 -2.096 0.036938 *
## sd1 1.171e+04 1.524e+03 7.679 2.45e-13 ***
## sd2 -6.633e+03 1.762e+03 -3.763 0.000203 ***
## sd3 2.153e+03 1.814e+03 1.187 0.236086
## sd4 -2.597e+03 1.980e+03 -1.312 0.190592
## sd5 1.484e+04 1.444e+03 10.273 < 2e-16 ***
## sd6 2.908e+03 1.768e+03 1.645 0.101035
## sd7 2.148e+03 1.813e+03 1.185 0.236967
## sd8 2.010e+03 1.859e+03 1.081 0.280471
## sd9 1.747e+02 1.544e+03 0.113 0.909996
## sd10 7.042e+03 1.527e+03 4.612 5.99e-06 ***
## sd11 3.875e+03 1.475e+03 2.627 0.009061 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 4618 on 291 degrees of freedom
## Multiple R-Squared: 0.7048, Adjusted R-squared: 0.6774
## F-statistic: 25.73 on 27 and 291 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation Massena:
## ========================================
## Massena = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11
##
## Estimate Std. Error t value Pr(>|t|)
## S_S_Marie.l1 -5.701e-03 9.792e-02 -0.058 0.9536
## Detroit.l1 5.101e-03 1.152e-02 0.443 0.6583
## Port_Huron.l1 1.752e-03 3.624e-02 0.048 0.9615
## B_Niagara.l1 3.003e-02 1.372e-02 2.189 0.0294 *
## Alexandria.l1 -9.495e-02 7.316e-02 -1.298 0.1954
## Ogdensburg.l1 2.417e-02 1.618e-01 0.149 0.8814
## Massena.l1 -4.036e-01 6.607e-02 -6.108 3.22e-09 ***
## Champlain_Point.l1 -8.234e-02 5.361e-02 -1.536 0.1256
## S_S_Marie.l2 5.973e-03 9.703e-02 0.062 0.9510
## Detroit.l2 1.773e-02 1.159e-02 1.530 0.1271
## Port_Huron.l2 5.462e-02 3.578e-02 1.527 0.1279
## B_Niagara.l2 -1.429e-02 1.397e-02 -1.023 0.3072
## Alexandria.l2 6.830e-02 7.418e-02 0.921 0.3579
## Ogdensburg.l2 6.485e-02 1.594e-01 0.407 0.6845
## Massena.l2 -4.029e-01 6.445e-02 -6.252 1.44e-09 ***
## Champlain_Point.l2 -1.016e-01 5.419e-02 -1.874 0.0619 .
## sd1 1.753e+04 3.072e+03 5.707 2.83e-08 ***
## sd2 -7.656e+03 3.551e+03 -2.156 0.0319 *
## sd3 4.805e+02 3.655e+03 0.131 0.8955
## sd4 -2.446e+03 3.990e+03 -0.613 0.5403
## sd5 2.442e+04 2.910e+03 8.392 2.11e-15 ***
## sd6 8.249e+03 3.562e+03 2.316 0.0213 *
## sd7 7.897e+03 3.653e+03 2.162 0.0314 *
## sd8 5.021e+03 3.745e+03 1.341 0.1811
## sd9 -3.815e+03 3.111e+03 -1.226 0.2211
## sd10 1.413e+04 3.077e+03 4.591 6.59e-06 ***
## sd11 2.986e+03 2.972e+03 1.005 0.3160
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 9305 on 291 degrees of freedom
## Multiple R-Squared: 0.6324, Adjusted R-squared: 0.5983
## F-statistic: 18.54 on 27 and 291 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation Champlain_Point:
## ================================================
## Champlain_Point = S_S_Marie.l1 + Detroit.l1 + Port_Huron.l1 + B_Niagara.l1 + Alexandria.l1 + Ogdensburg.l1 + Massena.l1 + Champlain_Point.l1 + S_S_Marie.l2 + Detroit.l2 + Port_Huron.l2 + B_Niagara.l2 + Alexandria.l2 + Ogdensburg.l2 + Massena.l2 + Champlain_Point.l2 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11
##
## Estimate Std. Error t value Pr(>|t|)
## S_S_Marie.l1 -9.240e-02 1.866e-01 -0.495 0.62096
## Detroit.l1 3.952e-02 2.196e-02 1.800 0.07293 .
## Port_Huron.l1 1.159e-02 6.908e-02 0.168 0.86688
## B_Niagara.l1 6.589e-03 2.615e-02 0.252 0.80122
## Alexandria.l1 -3.530e-03 1.395e-01 -0.025 0.97982
## Ogdensburg.l1 1.579e-01 3.085e-01 0.512 0.60924
## Massena.l1 7.874e-02 1.259e-01 0.625 0.53232
## Champlain_Point.l1 -5.268e-01 1.022e-01 -5.155 4.70e-07 ***
## S_S_Marie.l2 -7.096e-02 1.850e-01 -0.384 0.70150
## Detroit.l2 -3.005e-03 2.209e-02 -0.136 0.89188
## Port_Huron.l2 8.893e-02 6.820e-02 1.304 0.19323
## B_Niagara.l2 -1.557e-02 2.662e-02 -0.585 0.55917
## Alexandria.l2 5.524e-02 1.414e-01 0.391 0.69634
## Ogdensburg.l2 -1.282e-01 3.039e-01 -0.422 0.67340
## Massena.l2 -7.616e-02 1.228e-01 -0.620 0.53576
## Champlain_Point.l2 -4.302e-01 1.033e-01 -4.164 4.12e-05 ***
## sd1 7.085e+04 5.855e+03 12.099 < 2e-16 ***
## sd2 -4.108e+04 6.770e+03 -6.068 4.03e-09 ***
## sd3 3.188e+03 6.967e+03 0.458 0.64760
## sd4 -1.427e+04 7.606e+03 -1.876 0.06162 .
## sd5 8.857e+04 5.547e+03 15.968 < 2e-16 ***
## sd6 1.508e+04 6.790e+03 2.220 0.02718 *
## sd7 -3.988e+03 6.963e+03 -0.573 0.56731
## sd8 1.731e+03 7.139e+03 0.242 0.80859
## sd9 -6.274e+03 5.931e+03 -1.058 0.29104
## sd10 3.483e+04 5.866e+03 5.937 8.23e-09 ***
## sd11 1.807e+04 5.665e+03 3.189 0.00158 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 17740 on 291 degrees of freedom
## Multiple R-Squared: 0.8586, Adjusted R-squared: 0.8455
## F-statistic: 65.47 on 27 and 291 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## S_S_Marie Detroit Port_Huron B_Niagara Alexandria Ogdensburg
## S_S_Marie 98190338 3.504e+08 1.812e+08 7.428e+08 95236480 29015160
## Detroit 350355533 4.133e+09 1.140e+09 3.544e+09 423069383 123549616
## Port_Huron 181217837 1.140e+09 7.017e+08 1.900e+09 242475925 68027249
## B_Niagara 742758412 3.544e+09 1.900e+09 8.193e+09 991700687 256197746
## Alexandria 95236480 4.231e+08 2.425e+08 9.917e+08 172094772 35000465
## Ogdensburg 29015160 1.235e+08 6.803e+07 2.562e+08 35000465 21315968
## Massena 45281274 2.909e+08 1.286e+08 4.453e+08 59492373 13218819
## Champlain_Point 124389888 5.585e+08 3.388e+08 1.248e+09 168695814 55733975
## Massena Champlain_Point
## S_S_Marie 45281274 1.244e+08
## Detroit 290899826 5.585e+08
## Port_Huron 128583715 3.388e+08
## B_Niagara 445321246 1.248e+09
## Alexandria 59492373 1.687e+08
## Ogdensburg 13218819 5.573e+07
## Massena 86561809 7.278e+07
## Champlain_Point 72783064 3.146e+08
##
## Correlation matrix of residuals:
## S_S_Marie Detroit Port_Huron B_Niagara Alexandria Ogdensburg
## S_S_Marie 1.0000 0.5500 0.6904 0.8281 0.7326 0.6342
## Detroit 0.5500 1.0000 0.6695 0.6090 0.5016 0.4162
## Port_Huron 0.6904 0.6695 1.0000 0.7923 0.6978 0.5562
## B_Niagara 0.8281 0.6090 0.7923 1.0000 0.8352 0.6131
## Alexandria 0.7326 0.5016 0.6978 0.8352 1.0000 0.5779
## Ogdensburg 0.6342 0.4162 0.5562 0.6131 0.5779 1.0000
## Massena 0.4912 0.4863 0.5217 0.5288 0.4874 0.3077
## Champlain_Point 0.7077 0.4898 0.7211 0.7774 0.7250 0.6806
## Massena Champlain_Point
## S_S_Marie 0.4912 0.7077
## Detroit 0.4863 0.4898
## Port_Huron 0.5217 0.7211
## B_Niagara 0.5288 0.7774
## Alexandria 0.4874 0.7250
## Ogdensburg 0.3077 0.6806
## Massena 1.0000 0.4410
## Champlain_Point 0.4410 1.0000
# Residual diagnostics
serial.test(var)
##
## Portmanteau Test (asymptotic)
##
## data: Residuals of VAR object var
## Chi-squared = 1582.5, df = 896, p-value < 2.2e-16
resid.var <- resid(var)
plot.ts(resid.var,ylab = 'Residuals',xlab='Year')
# ACF of residuals
acf(resid(var), lag.max = 25)
# PACF of residuals
#pacf(resid.var)
pacf(resid(var))
#selecting the variables
# Granger test for causality
#for causality function to give reliable results we need all the variables of the multivariate time series to be stationary.
# Run the Granger causality tests for each cause variable
granger_results <- lapply(c("S_S_Marie", "Detroit", "Port_Huron", "B_Niagara",
"Alexandria", "Ogdensburg", "Massena", "Champlain_Point"),
function(cause_var) {
causality(var.a, cause = cause_var)
})
# Extract p-values for Granger causality and Instant tests for each port
p_values <- sapply(granger_results, function(result) {
c(Granger = result$Granger$p.value, Instant = result$Instant$p.value)
})
# Convert the p-values into a data frame for better readability
p_values_df <- as.data.frame(p_values)
colnames(p_values_df) <- c("S_S_Marie", "Detroit", "Port_Huron", "B_Niagara",
"Alexandria", "Ogdensburg", "Massena", "Champlain_Point")
# Print or further analyze the results
print(t(p_values_df))
## Granger Instant
## S_S_Marie 0.399929931 0.000000e+00
## Detroit 0.002567269 0.000000e+00
## Port_Huron 0.010626893 0.000000e+00
## B_Niagara 0.059133132 0.000000e+00
## Alexandria 0.317753694 0.000000e+00
## Ogdensburg 0.033622568 0.000000e+00
## Massena 0.954878920 3.701484e-13
## Champlain_Point 0.136128747 0.000000e+00
# Forecast next 5 steps
pred_var <- predict(var, n.ahead = 5, ci = 0.95)
pred_var
## $S_S_Marie
## fcst lower upper CI
## [1,] -20772.92 -40234.268 -1311.5627 19461.35
## [2,] 13959.33 -7154.523 35073.1802 21113.85
## [3,] 11920.56 -9822.885 33664.0029 21743.44
## [4,] -21830.58 -43945.540 284.3868 22114.96
## [5,] 26649.10 4490.964 48807.2387 22158.14
##
## $Detroit
## fcst lower upper CI
## [1,] -69030.416 -195086.91 57026.07 126056.5
## [2,] 38109.268 -95731.82 171950.35 133841.1
## [3,] -526.438 -135958.98 134906.11 135432.5
## [4,] -42202.056 -178590.67 94186.56 136388.6
## [5,] 66703.873 -69821.48 203229.22 136525.4
##
## $Port_Huron
## fcst lower upper CI
## [1,] -29916.707 -81837.54 22004.12 51920.83
## [2,] 29912.573 -26586.91 86412.06 56499.49
## [3,] 3831.647 -53965.93 61629.23 57797.58
## [4,] -34633.737 -93753.94 24486.47 59120.20
## [5,] 71254.895 12034.92 130474.87 59219.98
##
## $B_Niagara
## fcst lower upper CI
## [1,] -190069.68 -367515.73 -12623.62 177446.1
## [2,] 113009.35 -80055.64 306074.33 193065.0
## [3,] 67019.32 -131628.86 265667.50 198648.2
## [4,] -143410.26 -346587.43 59766.91 203177.2
## [5,] 311700.50 107992.62 515408.38 203707.9
##
## $Alexandria
## fcst lower upper CI
## [1,] -16351.510 -42063.35 9360.334 25711.84
## [2,] 16708.205 -12191.63 45608.040 28899.84
## [3,] 2860.326 -26517.35 32238.001 29377.68
## [4,] -21273.695 -51281.03 8733.640 30007.33
## [5,] 48076.143 17933.42 78218.868 30142.73
##
## $Ogdensburg
## fcst lower upper CI
## [1,] -7106.390 -16156.6616 1943.881 9050.271
## [2,] 4926.983 -4850.0801 14704.046 9777.063
## [3,] 2281.777 -7830.5937 12394.148 10112.371
## [4,] -6614.895 -16993.4306 3763.640 10378.536
## [5,] 10119.308 -284.3543 20522.971 10403.663
##
## $Massena
## fcst lower upper CI
## [1,] -15421.0121 -33659.028 2817.004 18238.02
## [2,] 11672.2780 -7934.703 31279.259 19606.98
## [3,] 714.6443 -19649.571 21078.859 20364.22
## [4,] -10464.0237 -31350.605 10422.558 20886.58
## [5,] 14522.1337 -6459.135 35503.402 20981.27
##
## $Champlain_Point
## fcst lower upper CI
## [1,] -38442.86 -73208.366 -3677.355 34765.51
## [2,] 28686.15 -9263.512 66635.818 37949.67
## [3,] 11710.02 -27729.883 51149.927 39439.91
## [4,] -33431.23 -74112.424 7249.957 40681.19
## [5,] 65000.35 24232.541 105768.150 40767.80
# plot the forecasts
par(mar=c(3,3,1,1))
plot(pred_var)
# Comparison with univariate AR models #
# time series length for out-of-sample forecasting
T0 <- 12
# length of training data set
T <- dim(multivariate_ts)[1] - T0
ts <- multivariate_ts[1:T,]
ts1 <- ts[-(1:9),1]
ts2 <- ts[-(1:9),2]
# univariate prediction
auto.arima(ts2,max.p=5,max.q=5,max.P = 0,max.Q=0,ic="aic",allowmean = TRUE)
## Series: ts2
## ARIMA(0,1,2)
##
## Coefficients:
## ma1 ma2
## -0.5245 -0.1642
## s.e. 0.0561 0.0541
##
## sigma^2 = 5.16e+09: log likelihood = -3766.99
## AIC=7539.97 AICc=7540.05 BIC=7551.07
auto.arima(ts1,max.p=5,max.q=5,max.P = 0,max.Q=0,ic="aic",allowmean = TRUE)
## Series: ts1
## ARIMA(1,1,5) with drift
##
## Coefficients:
## ar1 ma1 ma2 ma3 ma4 ma5 drift
## -0.1291 -0.8294 -0.1676 0.0153 0.1037 0.1407 -440.5315
## s.e. 0.3480 0.3430 0.3347 0.0772 0.0720 0.0891 194.4874
##
## sigma^2 = 213330552: log likelihood = -3288.92
## AIC=6593.85 AICc=6594.35 BIC=6623.45
#sarima(y,p=0,q=3,P=0,Q=1,S=4,d=0)
#ts1_ar <- sarima(ts1, p=1,q=1,P=0,Q=0,S=12,d=0, include.mean = TRUE, method = "ML")
#ts2_ar <- sarima(ts2, p=0,q=0,P=0,Q=0,S=12,d=0, include.mean = TRUE, method = "ML")
ts1_ar <- arima(ts1, order=c(1,0,1), include.mean = TRUE, method = "ML")
ts2_ar <- arima(ts2, order=c(0,0,0), include.mean = TRUE, method = "ML")
ts1.forecast <- predict(ts1_ar, T0)
ts2.forecast <- predict(ts2_ar, T0)
diff1 <- mean((ts1.forecast$pred[1:T0] - multivariate_ts[(T+1):(T+T0),1])^2)
diff2 <- mean((ts2.forecast$pred[1:T0] - multivariate_ts[(T+1):(T+T0),2])^2)
# Forecast error from univariate model
diff1
## [1] 280653543
diff2
## [1] 94427104252
diff1+diff2
## [1] 94707757795
# Prediction using multivariate model
var0 <- vars::VAR(ts[-(1:9),], p=2, type="both", season = 12)
pred_var0 <- predict(var0, n.ahead = T0, ci = 0.95)
# Forecast error from multivariate model
diff1_m <- mean((pred_var0$fcst$S_S_Marie[,1] - multivariate_ts[(T+1):(T+T0),1])^2)
diff2_m <- mean((pred_var0$fcst$Detroit[,1] - multivariate_ts[(T+1):(T+T0),2])^2)
diff1_m
## [1] 141455315
# Comparison with univariate ARIMA models #
# time series length for out-of-sample forecasting
T0 <- 12
# length of training data set
T <- dim(stnry_ts)[1] - T0
ts <- stnry_ts[1:T, ]
ts1 <- ts[-(1:9), 1]
ts2 <- ts[-(1:9), 2]
# Univariate ARIMA model for ts1
arima_ts1 <- auto.arima(ts1, max.p = 5, max.q = 5, max.P = 0, max.Q = 0, ic = "aic", allowmean = TRUE)
ts1_ar <- arima(ts1, order = arima_ts1$arma[c(1, 1, 5)], include.mean = TRUE, method = "ML")
# Univariate ARIMA model for ts2
arima_ts2 <- auto.arima(ts2, max.p = 5, max.q = 5, max.P = 0, max.Q = 0, ic = "aic", allowmean = TRUE)
ts2_ar <- arima(ts2, order = arima_ts2$arma[c(1, 1, 2)], include.mean = TRUE, method = "ML")
# Forecast using ARIMA models
ts1_forecast <- predict(ts1_ar, n.ahead = T0)
ts2_forecast <- predict(ts2_ar, n.ahead = T0)
# Calculate forecast errors for ARIMA models
diff1_ar <- mean((ts1_forecast$pred - stnry_ts[(T + 1):(T + T0), 1])^2)
diff2_ar <- mean((ts2_forecast$pred - stnry_ts[(T + 1):(T + T0), 2])^2)
# Multivariate VAR model
var0 <- vars::VAR(ts[-(1:9), ], p = 2, type = "both", season = 12)
pred_var0 <- predict(var0, n.ahead = T0, ci = 0.95)
# Forecast errors for VAR model
diff1_var <- mean((pred_var0$fcst$S_S_Marie[, 1] - stnry_ts[(T + 1):(T + T0), 1])^2)
diff2_var <- mean((pred_var0$fcst$Detroit[, 1] - stnry_ts[(T + 1):(T + T0), 2])^2)
diff1_var_multivariate <- mean((pred_var0$fcst$S_S_Marie[, 1] - stnry_ts[(T + 1):(T + T0), 1])^2)
# Create data frame for forecast errors
forecast_errors <- data.frame(
Model = c(rep("ARIMA Univariate", each = 3), rep("VAR Multivariate", each = 3)),
Series = rep(c("Yt_1", "Yt_2", "Yt_1 + Yt_2"), 2),
Forecast_Error = c(diff1_ar, diff2_ar, diff1_ar + diff2_ar, diff1_var, diff2_var, diff1_var + diff2_var)
)
# Print forecast errors table
print(forecast_errors)
## Model Series Forecast_Error
## 1 ARIMA Univariate Yt_1 262452430
## 2 ARIMA Univariate Yt_2 7777405896
## 3 ARIMA Univariate Yt_1 + Yt_2 8039858326
## 4 VAR Multivariate Yt_1 345681233
## 5 VAR Multivariate Yt_2 6057116931
## 6 VAR Multivariate Yt_1 + Yt_2 6402798165